Averaging Multiples: The Harmonic Mean

Suppose you're constructing a peer group of five stocks in order to create a benchmark P/E ratio. Four of them are exactly 15, and the fifth is 274, since trailing earnings were near zero for that firm. The arithmetic mean of 66.8 might not be the benchmark you're really looking for. Do you think this is too low or too high?

Of course.

No. This is far too high.

Clearly the "real" mean value is lower. Most are 15, so that's a tempting choice, but you'll want something that doesn't allow such a skew, leaving you with a much lower value.

You may recall considering the inverse of the P/E—which is the earnings yield, or the E/P. Since earnings is the value which can be zero or negative, just average the earnings yield instead. $$\displaystyle \frac{\frac{1}{15} + \frac{1}{15} + \frac{1}{15} + \frac{1}{15} + \frac{1}{274}}{5} = 0.0541 $$ That certainly is lower, but it's also not what you want. How can you get your benchmark P/E from here?

No. That would mix P/E ratios with E/P ratios. Since you already have an average E/P, just flip it over.

Absolutely. Turn that E/P into a P/E again.

The inverse of 0.0541 is $$\displaystyle \frac{1}{0.0541} \approx 18.5 $$. Much better. A reasonable value that still allows the effect of the higher observation. But it's lower than the arithmetic mean.

What you've really done here is calculate the __harmonic mean__, which is the inverse of the average inverse, if that makes sense. Here's the notation: $$\displaystyle X_H = \frac{n}{\sum_{i=1}^{n}\frac{1}{X_i}}$$. If you want to use weightings on the observations, that's fine, too. Then you'll use the __weighted harmonic mean__, which just adds a little omega as the weight with each observation. $$\displaystyle X_{WH} = \frac{1}{\sum_{i=1}^{n}\frac{\omega_i}{X_i}}$$

Since these calculations always place less emphasis on higher ratios, the harmonic mean will always be equal to or lower than the arithmetic mean. And the larger the dispersion of observations, the larger the difference between the two. What do you think this means in the case that you have a group of price multiples, such as several price-to-sales ratios, one ratio is far _lower_ than the rest of the group?

No. The harmonic mean would actually be further away from the rest of the group than the arithmetic mean.

That's right.

Same reason why: the harmonic mean is always lower. So be mindful of how that works. This also highlights why you need to know what "average EV/EBITDA" (or whatever ratio you're looking at) really means. It could be an arithmetic mean, weighted mean, harmonic mean, or weighted harmonic mean (that's what it probably will be). The weighted harmonic mean is probably the most common, using portfolio value weights in that calculation.

To summarize: [[summary]]

Too low

Too high

Flip it over

Average this with 66.8

The harmonic mean would be closer to the rest of the group than the arithmetic mean

The harmonic mean would be further away from the rest of the group than the arithmetic mean

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